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Einstein’s quantum oscillator in an equilibrium state:

Thermo-Field Dynamics or Statistical Mechanics?

Prof. A.Soukhanov,Russia, OGOL@OLDI. RU

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Setting of the problemAs it is known the Einstein’s initial theory of fluctuations covers only the

domain of thermal fluctuations at values of temperature T determined by an inequality

where kB – is the Boltzmann’s constant; τ – is a specific time interval for the system. For other values of temperatures T and intervals τ an essential role

play or quantum fluctuations (the subject of Quantum Mechanics) either they with thermal ones simultaneously. But now there is no successful theory joined both types fluctuations.

Meanwhile there is a great needs in the such theory. Open quantum systems are of interest. Among them are systems with a few number of freedom degrees by a low temperature and systems with unstable ground state. There are some reasons to suppose that quantum and thermal fluctuations are correlated among themselves and their values are comparable.

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The aims of our studies are

modification of normal thermodynamics andextension of Einstein’s fluctuations theory to the case

when thermal and quantum uncontrollable influences are enjoying equal right.

As a tool of our studies we use Schroedinger’s uncertainties relations (SHUR), which allow us to consider not only non-commutation but as well correlation of conjugated observables (characteristics) of system. They reflect themselves geometrical properties of the corresponding space and therefore their using assigns an universal meaning to description of Nature.

As a model of the open quantum system we use a quantum oscillator in the thermostat (thermal bath).

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Some important results

- an inconsequence of an approach in the frames of the Quantum Statistical Mechanics to description of quantum oscillator in the thermostat is established

- an advantages of an approach based on Thermo-Field Dynamics are shown

- a confirmation of existing of thermal noise in the pure thermal state is found

- a generalized definition of entropy is suggested- the canonical distribution is modified by means of

entering of generalized temperature- a program of generalization of Einstein’s fluctuations

theory and creating of the Quantum StatisticalThermodynamics is planned

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The main Hypothesis Taking as a model of an open system the oscillator we will assume that when ω – is a frequency of classical oscillator.We will represent thermostat as infinite set of sequences of N identical bound

quantum oscillators with frequencies in interval 0 ,where N . The Hypothesis: a quantum oscillator might be in thermal equilibrium state with the

thermostat at any temperature including T=0. In that case the energy of system will fluctuate even at T=0 due to a resonant binding between vacuum oscillators and the system.

As it is known the average value of energy of quantum oscillator in thermostat is

at high temperatures

at low temperatures For description unification let us suppose in this Formula at T0

where TW is the Wigner temperature (1932) contained in the Wigner function for quantum oscillator at T = 0 .

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Some special features of SHUR

We shall deal with oscillator in the thermal equilibrium state at T=0 to examine some special features of SHUR in this case.

For observables coordinate q and momentum p it has a form

, where

is the sum of the correlator square

andcommutator square

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Some calculations Calculating separately the left and right hand sides of SHUR at the

ground state one will find

Thus in this case at the condition SHUR takes the form of strong equality, named saturated SHUR.

Let us assume that the saturation of SCUR for momentum-coordinate is a property of thermal equilibrium state at any temperature even if σqp ≠ 0.

Note that there is a peculiar dependence between observables p and q even in this state as their dispersions are proportional to the Wigner temperature.

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A hypothesis verification by means of Quantum Statistical Mechanics

Suggesting that perhaps at any temperature SHUR for oscillator in

thermostat should be also a saturated Let us check this hypothesis We start with using of the usual Quantum Statistical

Mechanics in a mixed state. In this theory:the temperature of a system in thermal equilibrium

state is rigorously equal to the thermostat temperature;

coordinate and momentum of the system are considered as independent observables.

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In this case the coordinate and momentum dispersions are increasing together with temperature

But the right-hand part of SHUR does not change under heating, since

and

Nevertheless the dispersions q and p are proportional to each other but σqp is not dependent on the temperature. We focus attention on this

contradiction!Thus SHUR at T>0 stops to be saturated, and the Quantum Statistical

Mechanics becomes useless for solving of our problem, as expected.

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A hypothesis verification by means of Quantum Mechanics

Now let us try to use the Quantum Mechanics for our aim. It must be admitted that saturating of SHUR it is possible not only by the condition

. There exist a collection of pure states with for which SHUR is saturated. They are called correlated coherent

states (CCS) or squeezed states. Among these are all complex wave functions of Gaussian type.

where α is any function of time independent on coordinate.

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Obtaining a saturated SHUR At this state we obtain

The independent calculation of right-hand side of SHUR gives us

; As one can see we obtain just the same expression in the both sides The SHUR is saturated as expected !

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A verification by means of Thermo-Field Dynamics

Let us find among this states a wave function corresponding to the thermal equilibrium state. With another words we have to determine a concrete α

This problem might be successful solved by methods of Thermo-Field-Dynamics (TFD). This is a variant of Quantum Field Theory at finite temperature in real time created by Umezawa.

It assumes thermostat as well as vacuum are systems with infinite number of freedom degrees.

The main idea of TFD:Under the influence of thermostat appear two independent possibilities of energy

absorption by a system:-either an excitation of new quanta;-either a filling of new vacancies.Thus, putting of a system into the thermostat is equivalent to an effective doubling

of freedom degrees number. It results in cutting of a peculiar degeneration of state. Therefore we transfer from initial vacuum for particles |0> to a new vacuum for quasi-particles |0>>, which is dependent from temperature.

It is possible to do it by means u-v – transformation of Bogoljubov for a system with infinity number of freedom degrees.

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Obtaining of α

In the TFD framework it is possible to show that a pure thermal state in thermal equilibrium is fixed by value of α

Now the right-hand side of SHUR takes the form

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The analysis of TFD results for quantum oscillator (1)

A). From the saturated SHUR for observables p and q it follows

The quantity has a meaning of an effective quantum of action dependent on T. In particular at T>>TW we have

Then a statistical weight

We think of as the transition should be observed by experiment and

be an essential part of a new generalized theory like TFD at T0.

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The analysis of TFD results for quantum oscillator (2)

B). The obtained value of allows to say that

dependence of the wave function phase on temperature takes place. Namely

It means that we face with existence of "thermal noise" in a pure state. It remains to be valid at T>>TW .

C).This fact is contrary to the definition of entropy suggested by von Neumann. According it the entropy in the pure state must be equal to zero.

We propose to change this definition in order to account a contribute of phase dependent on temperature. In our opinion the generalized definition of entropy might have the following symmetric form:

The phase contribution is accounted indirectly for the second term in this sum.

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The analysis of TFD results for quantum oscillator (3)

Then for a system at the pure state in the thermostat with wave function

we obtain

A new essential feature of this quantity at T=0 (the Third Principle!)

where 0 ≡ e has a sense of the statistical weight at T=0. This is an universal quantity – it has the same value for any oscillators.

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A program for generalizing of Thermodynamics (1)

On the basis of these considerations we propose a program for generalizing of Thermodynamics – transition to Quantum Statistical Thermodynamics

In the traditional Quantum Statistical Mechanics one starts from the solving of quantum dynamics problem without considering of thermostat and then a system is placed into the thermostat.

We propose using an inverse consequence of actions: to start from the entering of thermo-field vacuum having the

temperature Tgen and then to put a system into it describing at the same time a system

behavior by classical means.A model of thermostat at the temperature T: an infinite set of independent sequences of N bounded quantum

oscillators having a frequency , where N is an infinite number and 0 .

Thereby a vacuum vector is an infinite product of independent vacuum vectors specific to each oscillator sequence. The system under study (if it can be approximated by quantum oscillator)

interacts by resonant means with a proper sequence of vacuum oscillators.

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A program for generalizing of Thermodynamics (2)

In all formulas of thermodynamics we propose to use the Planck average energy of quantum oscillator

as an alternative to the average energy of classical oscillator

Taking for unification of our description we obtain

where Tgen is the generalized temperature for the first time

referred by Bloch (1932).

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Generalization of formulas We think of as the generalization of thermodynamics may be carried out if we substitute for

generalized temperature Tgen instead T into the all formulas of usual thermodynamics:the generalized distribution of Gibbs

the generalized free energy

the generalized entropy

Let us note that the calculation of entropy gives us the same value as calculation by mean the formula for a pure state in the thermal equilibrium.

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Some results in the frames of this program (1)

Using the generalized canonical distribution (for mixed classical states) one can calculate the correlator of coordinate and momentum

By means of TFD (for pure states) we have obtained

Despite of Cgen =0 we see that

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Some results in the frames of this program (2)

II. The fluctuations of energy and temperature have the form

From this formulas we can see that even at T 0 the energy and the frequency of ground state have fluctuations

Thus the idea of the thermal state for the quantum oscillator gets an adequate description in this approach even at T=0

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Some results in the frames of this program (3)

III. Following to this way we suggest also the expression for generalized formula of Carno

Let us note that here TW is a common characteristic both for heater and for condenser because it is connected with the normal mode of vacuum oscillator which is equal to the frequency of the oscillator playing a role of operating substance.

At T2 ;T1 >> TW we get the usual formula

In conclusion may be said that there is real a difference between Quantum Statistical Mechanics and new Quantum Statistical Thermodynamics.

In our opinion using of generalized temperature open new possibilities for applications of the fluctuations theory to many problems in a sufficiently wide temperature domain and calculations of entropy for many objects.